The discrete groups acting nicely on the plane, called planar discontinous groups, are a classical topic, closely related to e.g. surface groups. Planar discontinous groups have Cayley graphs that can be embedded in the plane without accumulation points of vertices, like for example the familiar tessellations of the hyperbolic plain, or Escher’s tilings. There are however planar Cayley graphs all embeddings of which must have accumulation points of vertices. These graphs, and their groups, were far less well understood, and an open question due to Droms et. al. was whether they can be effectively enumerated.

We settle this question in the affirmative. The main part of our proof is to show that such groups admit nice presentations where the planarity ’can be seen’ in an automatic way.